030315 A Mixture of Generalized Negative Binomial ......The negative binomial distribution has...

J. Stat. Appl. Pro., 3 No. 3, 451-464 (2014) 451 Journal of Statistics Applications & Probability An International Journal

http://dx.doi.org/10.12785/jsap/030315

A Mixture of Generalized Negative Binomial Distribution

with Generalized Exponential Distribution

Adil Rashid*, Zahoor Ahmad and Tariq R Jan

P.G Department of Statistics, University of Kashmir, Srinagar, India

*Email:[email protected]

Received: 23 Mar. 2014, Revised: 14 Sep. 2014 Accepted: 16 Sep. 2014 Published: 1 Nov. 2014

ABSTRACT

The negative binomial distribution has become increasingly popular as a more flexible

alternative to Poisson distribution, especially when it is questionable whether the strict

requirements for Poisson distribution could be satisfied. But negative binomial distribution is

better for overdispersed count data that are not necessarily heavy-tailed, for heavy tailed count

data the traditional statistical distributions such as Poisson and negative binomial cannot be used

efficiently. In this paper an attempt has been made to obtain a mixture of generalized negative

binomial distribution with that of generalized exponential distribution, which is obtained by

mixing the generalized negative binomial distribution with generalized exponential distribution.

The new mixeddistribution so obtained generalizes several distributions that have been discussed

in literature. Estimation of the parameters, factorial moment and ordinary (crude) moments of the

new distribution has also been discussed. To justify the suitability, the distribution is fitted to a

reported count data set. The resulting fit is found to be good in comparison to others.

Keywords: Generalized negative binomial distribution, generalized exponential distribution,

mixture distribution and factorial moments.

http://dx.doi.org/10.12785/jsap/030315

mailto:[email protected]

452 A. Rashid et. al. A mixture of Generalized Negative…

1 Introduction

Traditional probability models such as Poisson, negative binomial distributions are

regarded as the most suitable models to represent count data. Poisson distribution is a standard

model for fitting count data when the number of occurrences of a phenomenon occurred at a

constant rate with respect to time and an occurrence of the changes of any future occurrences.

Equality of mean and variance is characteristic of the Poisson distribution but in a vast number of

practical applications, the count data are either over-dispersed (variance is greater than mean) or

under-dispersed (variance is less than mean) [12]. One frequent manifestation of over-dispersed

data is that the incidence of zero counts is greater than expected for the Poisson distribution and

this is of interests because zero counts frequently have special status, which is a violation of the

Poisson restrictions that the variance of the observed random variable equals its mean [18].

Statistical procedures must eliminate for these sources of variability in count data which

suggested a model in which the mean of Poisson distribution has a gamma distribution.

Practically this leads to the negative binomial distribution. The negative binomial distribution

has become increasingly popular as a more flexible alternative to Poisson distribution, especially

when it is questionable whether the strict requirements for Poisson distribution could be satisfied

[7] but negative binomial distribution is better for over-dispersed count data that are not

necessarily heavy-tailed.Wang [17] showed that extremely heavy tail implies over-dispersion but

the converse does not hold.Among the count data issues, which have a large number of zeros that

lead to heavy tail. For such data sets, the number of sites where no crash is observed is so large

that traditional statistical distributions or models, such as the Poisson and NB distributions,

cannot be used efficiently. Lord and Geedipall [9] showed that Poisson distribution tends to

underestimate the number of zeros given the mean of the data while the NB distributions may

overestimate zeros, but underestimate observations with a count.

The mixing of probability distributions is an important way to construct new probability

distributions that can be used as a more flexible alternative to traditional statistical distributions

especially under overdispersion. Based on the mixing mechanism various authors explored new

probability distributions for instance, Simon [16] obtained a negative binomial distribution by

mixing the mean of the Poisson distribution with gamma distribution. Panger and Willmot [11]

J. Stat. Appl. Pro., 3 No. 3, 451-464 (2014)/ www.naturalspublishing.com/Journals.asp 453

obtained a mixture of negative binomial distribution with exponential distribution and showed

the flexibility of this particular mixture in fitting count data. Deniz, Sarabia and Ojeda [5]

analyzed the univariate and multivariate versions of the mixture of negative binomial distribution

with inverse Gaussian distribution [6]. The mixture of Poisson distribution with Lindley for

modeling count data was introduced by Sankaran [15], later on Ghitany et.al [4] showed that in

many ways Lindley is a better distribution compared to exponential and hence it is natural, one

should expect a mixture of Poisson distribution with Lindley provides better fit compared to

Poisson-exponential. Zamani and Ismail [19] introduced a new mixed negative binomial by

mixing the negative binomial distribution with Lindley which has a thick tail and an alternative

for modeling count data of insurance claims which has a thick tail and a large value at zero. Adil

Rashid and Jan [13] obtained a compound of zero truncated generalized negative binomial

distribution with generalized beta distribution by ascribing a generalized beta distribution to

probability parameter. Aryuyuen and Bodhisuwan [1] obtained a mixture of negative binomial

distribution with a generalized exponential distribution which includes two more free parameters.

Most recently Adil Rashid and Jan [14] explored a new compound distribution by mixing Geeta

distribution with generalized beta distribution which contains several distributions as special

cases.

In this article, we introduce a new mixed generalized negative binomial distribution

,,m by mixing generalized negative binomial distribution with generalized exponential

distribution , where reparametrization of e is considered. This new mixed

distributions has a thick tail and may be considered as alternative for modeling count data of

insurance claims which has thick tail and large values of zeros.

2 Materials and methods

We begin by giving definitions and relations needed for mixing of probability distributions:

A discrete random variable is X said to have a generalized negative binomial distribution

(GNBD) with parameters m&, if its probability mass function is given by

,..1,0,)1(),,;(1

x

xxmx

x

xm

xm

mmxf

(1)


Where

)(.0,,10

)(,1;1,0,10

iiNm

im

For 1 equation (1) reduces to the negative binomial distribution (NBD).If

,0, forNm one obtains from (1) binomial distribution (BD) and for 1 , Pascal

distribution see Johnson, Kotz and Kemp [7].

In GNBD the parameters mand , are constants but here we have considered a problem

in which the probability parameter e where > 0 is itself a random variablefollowing

generalizedexponential distribution (GED) with parameters and and has probability density

function

)2(0,0;)1(),;( 1

2 forxeef xx

The GED was introduced by Gupta & Kundu [6]. The moment generating function

(m.g.f) of (2) given by

1

1)1(

),;()( 20

t

t

xfetM xtX

(3)

In order to obtain a mixture of GNBD with that of GED we first provide a general

definition of this distribution which will subsequently expose its probability mass function.

3 Mixture of GNBD with GED

Let |X be a random variable following GNBD ( em ,, ) where is distributed

as GED with positive parameters ( , ) i.e. ),(~&),,(~| GEDemGNBDX ,

then determining the distribution of X that results from marginalizing over is known as a

mixture of GNBD with GED.


Theorem3.1: The probability mass function of a mixture of generalized negative binomial

distribution (GNBD) with generalized exponential distribution (GED) is given by the expression

1

1)1(

)1(),,,;(0

xxmj

xxmj

j

x

x

xm

xm

mmxf j

x

j

(4)

where 0,,,;,...1,0 mx .

Proof: If ),,(~| emGNBDX and ),(~ GED , then the probability mass function of X

can be obtained by

0

21 ),;()|(),,,;( dfXfmxf (5)

Where )|(1 Xf is defined by

)(

1 )1()|( xxmx eex

xm

xm

mXf

)(

0

)1( xxmjjx

j

eej

x

x

xm

xm

m

)(

0

)1( xxmjjx

j

ej

x

x

xm

xm

m

(6)

Substituting (6) into (5) we obtain

0

2

)(

0

),;()1(),,,;(

dfe

j

x

x

xm

xm

mmxf xxmjj

x

j

)()1(0

xxmjMj

x

x

xm

xm

m jx

j

Using the argument (3) we get

1

1)1(

)1(0

xxmj

xxmj

j

x

x

xm

xm

m jx

j

Equivalently it can be written in the form


)7(,

)(

)()1(),,,;(

0

xxmjB

xxmj

xxmj

j

x

x

xm

xm

mmxf j

x

j

Where B(.) refers to the beta function

0,;)(

),(

sr

sr

srsrB

Special Cases:

Case (i): For 1 GNBD reduces to NBD, therefore the mixture of NBD with GED is followed

from (7) by simply substituting 1 in it.

)8(,

)(

)()1(

1),,;(

0

mjB

mj

mj

j

x

x

xmmxf j

x

j

the mixture distribution displayed in (8) was introduced by Sirinapa Aryuyuen et.al [1].

Case (ii): For Nm &0 GNBD reduces to BD, therefore a mixture of BD with GED is

followed from (8) by simply substituting 0 in it.

xmjB

xmj

xmj

j

x

x

mmxf j

x

j

,)(

)()1(),,;(

0

Case (iii): For 1 , GED reduces to exponential distribution (ED) and a mixture of GNBD with

ED is followed from (7) by simply substituting 1 in it.

xxmjB

xxmj

xxmj

j

x

x

xm

xm

mmxf j

x

j

,1)(

()1(),,;(

0

Case (4): For 0,1 m and 1 in (7) we obtain mixture of NBD with ED and its pmf is given

as

mjB

mj

mj

j

x

x

xmmxf j

x

j

,1)(

)()1(

1),;(

0

)9())((

1)1(1

1

0

jm

j

x

x

xmj

x

j

The mixture distribution displayed in (9) was introduced by Panger and Willmot [11].

Case (5): For Nm ,0 and 1 in (7) we obtain mixture of BD with ED, therefore, its pmf is

given by the expression


xmjB

xmj

xmj

j

x

x

mmxf j

x

j

,1)(

)()1(),;(

0

4 Factorial moments and ordinary (crude) moments of a mixture of negative binomial

distribution with generalized exponential distribution

If ~| X NBD ),( em , where ~ GED ( , ) then when one keeps in mind the so called

factorial polynomial

)1)...(2)(1( lxxxxx l

)10(|)( xExm ll

is called the factorial moment of order l of a mixture of NBD with GED, where |xl is the

factorial moment of NBD.

Theorem 4.1: The factorial moment of order l of a mixture of negative binomial distribution

with generalized exponential distribution is given by the expression

l

j

j

ljl

jl

j

l

m

lmxm

01

1)1(

)1()(

)(

where 0,...,2,1,0 andmforx

Proof: The factorial moment of order l of NBD ,m is

l

l

lm

lmx

)(

)1)((

Thus the factorial moment of order l of a mixture of NBD with GED is obtained from (10), if we

let e

l

l

l

l eEm

lm

em

elmExm )1(

)(

)(

)(

)1)(()(

Using the binomial expansion of le )1( we have

)(

0

)1()(

)( jljl

j

eEj

l

m

lm


)11()1()(

)()(

0

jlMj

l

m

lmxm j

l

j

l

From the moment generating function of GED (3) with jlt , we get

)12(

1

1)1(

)1()(

)()(

0

jl

jl

j

l

m

lmxm j

l

j

l

Special case:

If 1 in (12) we get the factorial moment of order l of a mixture of NBD with ED

jl

jl

j

l

m

lmxm j

l

j

l

2

1)2(

)1()(

)()(

0

1

0

1)1()(

)()(

jl

j

l

m

lmxm j

l

j

l

The ordinary (crude) moments of mixture distributions under considerations are obtained by

using the formula

j

l

J

l

jl mm S

0

where Sl

jstands for the so called Sterling numbers of the second kind. Bohlmann [2] seems to be

the first to give this formula; the tables for these numbers can be found, for instance, in

Lukasiewicz and Warmus [10].

5 Parameter estimation

In this section the estimation of parameters of mixture of GNBD with GED will be

discussed by using maximum likelihood estimation. The likelihood function ofa mixture of

GNBD with GED ),,,( m is given by

x

j

jn

i xxmj

xxmj

j

x

x

xm

xm

mmxL

011

1)1(

)1(),,,,(


n

i ii

ii

jx

j

i

iii

in

i

n

ii

xxmj

xxmj

j

x

xxmx

xm

xm

mmxLmx

i

1 0

1 1

1

11

)1(log

)1()1(

)1(loglog),,,,(log),,,,(£

(13)

1

11

)1(log

)1(log)1(log)1(log)log(log

1 0

11

n

i ii

ii

jx

j

i

n

i

iiii

n

i

i

xxmj

xxmj

j

x

xxmxxmxmm

i

The first order conditions for finding the optimal values of the parameters obtained by

differentiating (13) with respect andm ,, gives rise to the following differential equations

n

i

ii

ii

jx

j

i

ii

ii

jx

j

i

n

i ii

ii

i

in

i i

xxmj

xxmj

j

x

xxmj

xxmj

mj

x

xxm

xxmm

xm

xmm

xmmm

i

i

1

0

0

11

1

1

)1(

1

1

)1(

)1(

)1(

)1(

)1(11£


n

i

ii

ii

jx

j

i

ii

ii

jx

j

i

n

i

ii

n

i

i

n

i i

i

xxmj

xxmj

j

x

xxmj

xxmj

mj

x

xxmxmxmm

x

i

i

1

0

0

111

1

1

)1(

1

1

)1(

)1()1()(

(14)

where)(

)()(

i

ii

x

xx

is a digamma function.

n

i

ii

ii

jx

j

i

ii

ii

jx

j

i

ii

iin

i i

in

i i

i

xxmj

xxmj

j

x

xxmj

xxmj

j

x

xxm

xxm

xm

xm

xm

x

i

i

1

0

0

11

1

1

)1(

1

1

)1(

)1(

)1(

)1(

)1(

)(

£


n

i

ii

ii

jx

j

i

ii

ii

jx

j

i

n

i

ii

n

i

i

n

i i

i

xxmj

xxmj

j

x

xxmj

xxmj

j

x

xxmxmxm

x

i

i

1

0

0

111

1

1

)1(

1

1

)1(

)1()1()(

£

(15)

n

i

ii

jx

j

i

ii

jx

j

i

xxmjj

x

xxmjj

x

i

i

1

0

0

1

)1()1(

1

)1()1(

(16)

n

i

ii

ii

jx

j

i

ii

ii

jx

j

i

xxmj

xxmj

j

x

xxmj

xxmj

j

x

i

i

1

0

0

1

1

)1(

1

1

)1(

£

(17)

The above equations have difficult and complicated algebraic solution therefore we use Newton-

Raphson method which is a powerful technique for solving equations numerically. Like so much

of the differential calculus, it is based on the simple idea of linear approximation [3]. Thus we

estimated the parameters of distribution by equating (14-17) to zero, the MLE solution of


ˆˆ,ˆ,ˆ andm can be obtained by solving the resulting equations simultaneously using a

numerical procedure with Newton-Raphson method.

6 Results and Discussion

Here, we illustrate the applicability of the new proposed mixture distribution using a real data

set, the data contains number of automobile liability policies in Switzerland for private cars taken

form Klugman [8]. Normally these data sets are fitted by Poisson distribution, NB distribution

and NB-GE distribution. The maximum likelihood method provides parameter estimation. By

comparing these fitting distributions in table 1, based on the P value it is clear that a new mixture

distribution of GNBD with GED provides better fit for the count data that have a large number of

zeros.

Table:1 Observed and expected frequency of accident data

Number of

accidents Observed Frequency

Fitted Distribution

Poisson NBD NBD-GED GNBD-GED

0 103704 102633.7 103723.6 103708.8 103706.4

1 14075 15918.5 13989.9 14046.8 14068.6

2 1766 1234.5 1857.1 1797.8 1775.2

3 255

66.5

245.2 251.7 253.1

4 45

37.2

36 40.6

5 6

12 9.1

6 2

7 0

Total 119853 119853 119853 119853 119853

Parameter Estimation

0123.33ˆ

1333.169996.30ˆ

0139.5ˆ3010.3ˆ1509.

4193.3ˆ4132.2ˆ062.1ˆ1551.

mmm

Chi-square 1332.3 12.12 4.26 0.675

d.f. 2 2 2 2

p-value <0.001 0.0023 0.1188 0.714

7 Conclusion

In this paper we have introduced a new mixture distribution by mixing the GNBD with

GED known as a mixture of GNBD with GED,which contains several mixture distributions as its

special cases. Further, the parameter estimation, factorial moments and ordinary (crude)


moments of the new mixed distribution has also been obtained. Finally we described the

usefulness of the new proposed mixed distribution to the count data sets characterized by large

number of zeros.

References:

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Distribution, Applied Mathematical Sciences, 7(22), 1093-1105 (2013).

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Statistik, Math. Ann. 74, 341-409 (1913).

[3] M.D Bressoud, A Radical Approach to Real Analysis, 2nd

Edn, Cambridge Univ Press,

Cambridge, (2005).

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(2000)

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Distribution, Journal of Modern Applied Statistical Methods 13 (1), 278-286 (2014).

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Casualty Actuarial Society 17, 45–53 (1961).

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030315 A Mixture of Generalized Negative Binomial ......The negative binomial distribution has...

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